In this work, we explore the construction of topological spaces related to finite sets through graph-theoretical representations derived from conjugation relations. This study establishes a conceptual link between three fundamental areas of mathematics: set theory, graph theory, and topology. We introduce a framework, referred to as the "bridge between groups, graphs, and topology," in which a finite set is represented by a graph whose vertices correspond to the elements of a group, and whose edges are defined by conjugation relations. Using the adjacency matrix of the resulting graph, we derive an adjacency rule that constructs a topology on the set of vertices. This methodology facilitates the exploration of various topological properties, including continuity, open and closed subsets, and possible covering spaces. Furthermore, we present algorithmic procedures applicable within the GAP (Group, Algorithm and Programming) framework for the systematic and accurate construction and analysis of these graphs. The proposed approach is illustrated through several illustrative examples involving finite groups, particularly Dihedral groups and Klein four-group, demonstrating how distinct topologies arise naturally and reflect the underlying algebraic structure of the groups. These results contribute to strengthening the interaction between algebra and topology through graphical techniques, providing a unified framework with potential applications to broader classes of finite groups and graphs structure analysis. Furthermore, we introduce a new binary relation on finite groups, called the factorial relation. This relation is precisely defined, and its associated graph is constructed and studied. We prove that the topology induced by the factorial relation is always a discrete topology on the set of elements of the underlying group. This result confirms a strong correspondence between the algebraic properties of the factorial relation and the resulting topological structure.